\[ \int \frac {\sqrt {-1+x}}{\left (1+x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {x \sqrt {-1+x}}{4 \left (x^{2}+1\right )^{2}}-\frac {\left (1-11 x \right ) \sqrt {-1+x}}{32 \left (x^{2}+1\right )}-\frac {\arctan \left (\frac {-2 \sqrt {-1+x}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \sqrt {-1054+746 \sqrt {2}}}{128}+\frac {\arctan \left (\frac {2 \sqrt {-1+x}+\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right ) \sqrt {-1054+746 \sqrt {2}}}{128}-\frac {\ln \left (1-x -\sqrt {2}-\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {1054+746 \sqrt {2}}}{256}+\frac {\ln \left (1-x -\sqrt {2}+\sqrt {-1+x}\, \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {1054+746 \sqrt {2}}}{256} \]
command
integrate((-1+x)**(1/2)/(x**2+1)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________